Unproven claim in Weinberg's textbook

• dEdt
In summary, Weinberg claims that a solution to the Schrodinger equation for a central potential must have the form \psi(\mathbf{x})=R(r)Y(\theta,\phi) where R(r) is a function of r and Y(\theta,\phi) is a function of the spherical angles. This is because the operator \mathbf{L}^2 only acts on the angular part of the wavefunction, making it proportional to a function only of angles. He does not provide a proof for this statement and suggests working on the intuition behind it instead.

dEdt

In his Lectures on Quantum Mechanics, Weinberg makes the following claim about solutions to the Schrodinger equation for a central potential: Suppose $\psi(\mathbf{x},t)$ is an eigenfunction of $H$, $\mathbf{L}^2$, and $L_z$. According to Weinberg, "since $\mathbf{L}^2$ acts only on angles, such a wavefunction must be proportional to a function only of angles, with a coefficient of proportionality $R$ that can depend only on $r$. That is, for all r,
$$\psi(\mathbf{x})=R(r)Y(\theta,\phi)."$$

He does not elaborate further on this, in my view, non-trivial statement. Can someone here provide a proof of his claim?

dEdt said:
He does not elaborate further on this, in my view, non-trivial statement. Can someone here provide a proof of his claim?

I'd be inclined to work on the intuition behind that claim instead of looking for a rigorous proof, because the general idea is useful in so many other problems.

But first, which part of the statement are you asking about? The part about ##L^2## acting only on ##\theta## and ##\phi##, or the claim that this implies the ##R(r)Y(\theta,\phi)## form of the solution?

I'm talking about the claim that $\psi=R(r)Y(\theta,\phi)$ given that $\mathbf{L}^2$ acts only on angles.

This is trivial once you consider the uniparticle Hilbert space for the 'dummy' particle 'moving' aroung the CoM. Its Hilbert space is L^2(R^3) = L^2 (R) X L^2 (R) X L^2 (R) and after performing a unitary transformation of the whole space you have that L^2(R^3) ~ L^((0,infinity),dr) X L^2 (S^2,dOmega).

In the position space the wavefunction of the 'dummy' particle would be then a product of a (sq integrable) function of r times a product of a function of spherical angles.

As a fellow scientist, I understand the importance of critically evaluating claims made in scientific literature. In this case, Weinberg's claim may seem non-trivial and lacking in explanation, but it is actually a well-established result in quantum mechanics.

The Schrodinger equation for a central potential is a well-known problem in quantum mechanics, and the solutions to this equation are given by the eigenfunctions of the Hamiltonian, \mathbf{L}^2, and L_z. These eigenfunctions are often referred to as "spherical harmonics," denoted by Y(\theta,\phi), which are functions only of the angles \theta and \phi. This is because the operators \mathbf{L}^2 and L_z only act on the angular variables, leaving the radial variable r unchanged.

Therefore, the wavefunction \psi(\mathbf{x}) can be written as a product of a radial function R(r) and the spherical harmonic Y(\theta,\phi), as stated by Weinberg. This is a consequence of the fact that the Schrodinger equation is separable in spherical coordinates.

In conclusion, Weinberg's claim is a well-established result in quantum mechanics and does not require further proof. However, if you would like to learn more about the mathematical derivation of this result, I recommend referring to a textbook on quantum mechanics or consulting with a quantum physicist.

1. What is an unproven claim in Weinberg's textbook?

An unproven claim in Weinberg's textbook refers to a statement or assertion made by the author that has not been supported by sufficient evidence or scientific research.

2. How do unproven claims affect the credibility of Weinberg's textbook?

Unproven claims can significantly impact the credibility of Weinberg's textbook as they may be seen as unreliable or lacking in scientific rigor. This can make readers question the accuracy and validity of the information presented in the textbook.

3. Why would Weinberg include unproven claims in the textbook?

There could be several reasons why Weinberg includes unproven claims in the textbook. It could be due to a lack of available evidence at the time of writing, personal bias or opinion, or to stimulate further research and discussion on the topic.

4. How can readers identify unproven claims in Weinberg's textbook?

Readers can identify unproven claims by looking for statements that are not supported by evidence or scientific research. They can also look for phrases such as "it is believed" or "some researchers suggest," which indicate a lack of concrete evidence.

5. What should readers do when they encounter an unproven claim in Weinberg's textbook?

When encountering an unproven claim in Weinberg's textbook, readers should approach it with a critical and questioning mindset. They can research the claim further and look for supporting evidence from reputable sources. If no evidence is found, it is best to treat the claim with skepticism and seek out alternative sources for information on the topic.